A057538 Birthday set of order 5: numbers congruent to +-1 modulo 2, 3, 4 and 5.

1, 11, 19, 29, 31, 41, 49, 59, 61, 71, 79, 89, 91, 101, 109, 119, 121, 131, 139, 149, 151, 161, 169, 179, 181, 191, 199, 209, 211, 221, 229, 239, 241, 251, 259, 269, 271, 281, 289, 299, 301, 311, 319, 329, 331, 341, 349, 359, 361, 371, 379, 389, 391, 401, 409

Offset: 1

Author

Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000

Keywords

• easy
• nonn

Comments

Also numbers congruent to +-1 or +-11 modulo 30 and numbers k where (k^2 - 1)/120 is an integer; all but the first two prime legs of Pythagorean triangles which also have prime hypotenuses appear within in this sequence (A048161). - Henry Bottomley, Jan 31 2002

Numbers k such that k^2 == 1 (mod 30). - Gary Detlefs, Apr 16 2012

Subsequence of primes gives A045468. - Ray Chandler, Jul 29 2019

Examples

			229 is congruent to 1 (mod 2), 1 (mod 3), 1 (mod 4) and -1 (mod 5).
x+ 11*x^2 + 19*x^3 + 29*x^4 + 31*x^5 + 41*x^6 + 49*x^7 + 59*x^8 + 61*x^9 + ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• A. Feist, On the Density of Birthday Sets, The Pentagon, 60 (No. 1, Fall 2000), 31-35.
• A. Feist, Maple source for birthday sets. [Broken link]
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A007310, A045468, A057539, A057540, A057541, A093722.

Programs

Maple

for n from 1 to 409 do if (n^2 mod 30 =1) then print(n) fi od; # Gary Detlefs, Apr 17 2012

Mathematica

a057538[n_] := Block[{f},
  f[x_] :=
   If[Mod[x, #] == 1 || Mod[x, #] == # - 1, True, False] & /@
    Range[2, 5];
Select[Range[n], DeleteDuplicates[f[#]] == {True} &]]; a057538[409] (* Michael De Vlieger, Dec 26 2014 *)

PARI

{a(n+1) = (n\4*3 + n%4)*10 + (-1)^(n\2)} /* Michael Somos, Oct 17 2006 */

Formula

A093722(n) = (a(n)^2 - 1)/120.

G.f.: x * (1 + 10*x + 8*x^2 + 10*x^3 + x^4) / ((1 - x) * (1 - x^4)). a(-1 - n) = -a(n). - Michael Somos, Jan 21 2012

4*a(n) = 30*(n+1) - 45 + 5*(-1)^n + 6*(-1)^floor((n+1)/2). - R. J. Mathar, Jul 30 2019

Extensions

Corrected by Henry Bottomley, Jan 31 2002

Offset corrected to 1 by Ray Chandler, Jul 29 2019